Multiple periodic solutions of state-dependent threshold delay equations

Abstract

We prove the existence of multiple periodic solutions for scalar-valued state-dependent delay equations of the form $x'(t) = f(x(t - d(x_t)))$, where $d(x_t)$ is given by a threshold condition and $f$ is close, in a suitable sense, to the step function $h(x) = -\mbox{sign}(x)$. We construct maps whose fixed points correspond to periodic solutions and show that these maps have nontrivial fixed points via homotopy to constant maps.  &nbsp We also describe part of the global dynamics of the model equation $x'(t) = h(x(t - d(x_t)))$.